L(s) = 1 | + 3.00·3-s + 2.74·5-s − 3.93·7-s + 6.00·9-s + 2.93·11-s − 2.73·13-s + 8.25·15-s + 0.166·17-s + 0.325·19-s − 11.8·21-s − 0.107·23-s + 2.56·25-s + 9.00·27-s + 7.02·29-s − 4.61·31-s + 8.80·33-s − 10.8·35-s − 4.53·37-s − 8.19·39-s − 5.07·41-s + 12.0·43-s + 16.5·45-s + 5.57·47-s + 8.47·49-s + 0.500·51-s + 7.42·53-s + 8.06·55-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1.22·5-s − 1.48·7-s + 2.00·9-s + 0.884·11-s − 0.757·13-s + 2.13·15-s + 0.0404·17-s + 0.0747·19-s − 2.57·21-s − 0.0224·23-s + 0.512·25-s + 1.73·27-s + 1.30·29-s − 0.829·31-s + 1.53·33-s − 1.82·35-s − 0.746·37-s − 1.31·39-s − 0.791·41-s + 1.83·43-s + 2.46·45-s + 0.812·47-s + 1.21·49-s + 0.0701·51-s + 1.01·53-s + 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.701865936\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.701865936\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 3.00T + 3T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 0.166T + 17T^{2} \) |
| 19 | \( 1 - 0.325T + 19T^{2} \) |
| 23 | \( 1 + 0.107T + 23T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 - 7.42T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 2.67T + 73T^{2} \) |
| 79 | \( 1 - 7.51T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 6.98T + 89T^{2} \) |
| 97 | \( 1 - 2.36T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.894374332599659696044782219659, −7.01879472234922920209923551247, −6.69151522161751317684672041569, −5.89741641180516307330169370715, −5.00111421528144537785709534459, −3.84694248017402544119533198437, −3.50467366225167718177035867939, −2.42437658541514324985880725965, −2.28272474108637848869499773990, −1.00430442033054419815484532660,
1.00430442033054419815484532660, 2.28272474108637848869499773990, 2.42437658541514324985880725965, 3.50467366225167718177035867939, 3.84694248017402544119533198437, 5.00111421528144537785709534459, 5.89741641180516307330169370715, 6.69151522161751317684672041569, 7.01879472234922920209923551247, 7.894374332599659696044782219659